Sidney I. Resnick. A Probability Path. Birkhauser Boston Basel Berlin

Transcription

1 Sidney I. Resnick A Probability Path Birkhauser Boston Basel Berlin

2 Preface xi 1 Sets and Events Introduction Basic Set Theory Indicator functions Limits of Sets Monotone Sequences Set Operations and Closure Examples The a -field Generated by a Given Class C Borel Sets on the Real Line Comparing Borel Sets Exercises 20 2 Probability Spaces Basic Definitions and Properties More on Closure Dynkin's theorem Proof of Dynkin's theorem Two Constructions Constructions of Probability Spaces General Construction of a Probability Model Proof of the Second Extension Theorem 49

3 vi 2.5 Measure Constructions Lebesgue Measure on (0,1] Construction of a Probability Measure on E with Given Distribution Function F(x) Exercises 63 3 Random Variables, Elements, and Measurable Maps Inverse Maps Measurable Maps, Random Elements, Induced Probability Measures Composition Random Elements of Metric Spaces Measurability and Continuity Measurability and Limits a -Fields Generated by Maps Exercises 85 4 Independence Basic Definitions Independent Random Variables Two Examples of Independence Records, Ranks, Renyi Theorem Dyadic Expansions of Uniform Random Numbers More on Independence: Groupings Independence, Zero-One Laws, Borel-Cantelli Lemma Borel-Cantelli Lemma Borel Zero-One Law Kolmogorov Zero-One Law Exercises Integration and Expectation Preparation for Integration Simple Functions Measurability and Simple Functions Expectation and Integration Expectation of Simple Functions Extension of the Definition Basic Properties of Expectation Limits and Integrals Indefinite Integrals The Transformation Theorem and Densities Expectation is Always an Integral on R Densities The Riemann vs Lebesgue Integral Product Spaces 143

4 vii 5.8 Probability Measures on Product Spaces Fubini's theorem Exercises 155 Convergence Concepts Almost Sure Convergence Convergence in Probability Statistical Terminology Connections Between a.s. and i.p. Convergence Quantile Estimation L p Convergence Uniform Integrability Interlude: A Review of Inequalities More on L p Convergence Exercises 195 Laws of Large Numbers and Sums of Independent Random Variables Truncation and Equivalence A General Weak Law of Large Numbers Almost Sure Convergence of Sums of Independent Random Variables Strong Laws of Large Numbers Two Examples The Strong Law of Large Numbers for IID Sequences Two Applications of the SLLN The Kolmogorov Three Series Theorem Necessity of the Kolmogorov Three Series Theorem Exercises 234 Convergence in Distribution Basic Definitions Scheffe's lemma Scheffe's lemma and Order Statistics The Baby Skorohod Theorem The Delta Method Weak Convergence Equivalences; Portmanteau Theorem More Relations Among Modes of Convergence New Convergences from Old Example: The Central Limit Theorem for m-dependent Random Variables The Convergence to Types Theorem Application of Convergence to Types: Limit Distributions for Extremes Exercises 282

5 viii 9 Characteristic Functions and the Central Limit Theorem Review of Moment Generating Functions and the Central Limit Theorem Characteristic Functions: Definition and First Properties Expansions Expansion of e ix Moments and Derivatives Two Big Theorems: Uniqueness and Continuity The Selection Theorem, Tightness, and Prohorov's theorem The Selection Theorem Tightness, Relative Compactness, and Prohorov's theorem Proof of the Continuity Theorem The Classical CLT for iid Random Variables The Lindeberg-Feller CLT Exercises Martingales Prelude to Conditional Expectation: The Radon-Nikodym Theorem Definition of Conditional Expectation Properties of Conditional Expectation Martingales Examples of Martingales Connections between Martingales and Submartingales Doob's Decomposition Stopping Times Positive Super Martingales Operations on Supermartingales Upcrossings Boundedness Properties Convergence of Positive Super Martingales Closure Stopping Supermartingales Examples Gambler's Ruin Branching Processes Some Differentiation Theory Martingale and Submartingale Convergence Krickeberg Decomposition Doob's (Sub)martingale Convergence Theorem Regularity aijd Closure Regularity and Stopping Stopping Theorems 392

6 ix Wald's Identity and Random Walks The Basic Martingales Regular Stopping Times Examples of Integrable Stopping Times The Simple Random Walk Reversed Martingales Fundamental Theorems of Mathematical Finance A Simple Market Model Admissible Strategies and Arbitrage Arbitrage and Martingales Complete Markets Option Pricing Exercises 429 References 443 Index 445